Textbook file for the Fall 2013
download HERE

Second version HERE

   
     

 

 

 





INTRODUCTION TO DIFFERENTIAL GEOMETRY
MATH 448/548


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Class Hours: Tuesday, Thursday 7:20 pm – 8:40 pm in Pray-Harrold room 321

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Office Hours: Tuesday, Thursday:6:45– 7:20 pm

A. Catalog description:
Curve and surface theory in three-dimensional space: introduction to special and general relativity.

B. Pre-Requisite:
Math 223(Multivariable calculus), Math 325(Ordinary Differential equations)

C. Course Objectives:
1. develop your understanding of geometric concepts of ‘‘local” and “global’’ for curves and surfaces.
2. deal with intrinsic and extrinsic concepts of a three-dimensional surface.
3. master the concept of curvature for plane and space curves.
4. distinguish between different concepts of curvature of a surface and to understand their physical significance..
5. be able to use MAPLE package to compute geometric concepts.
6. Provide students with tools to produce publishable research in the field of Differential Geometry.

D. Usual Course Content:
Unit1: Introduction to the theory of plane and space curves.
Tangent vectors, normal, curvature, torsion, Frenet-Serret apparatus.
Unit 2: Introduction to the theory of surfaces; tangent plane, normal vector, first and second fundamental forms, Gauss’ formula, Christoffel symbols, Gaussian and mean curvature, Riemannian tensor of curvature, directional derivative, geodesic curves.
Unit 3. Special topics of theory of surfaces: Einstein surfaces, umbilical surfaces, minimal surfaces, surfaces of revolution, surfaces with constant curvature.
Unit 4. Introduction to the theory of manifolds: vector fields, Riemann and Lorentz metric, sectional curvatures, Schur’s theorem, linear connections and their curvature and torsion, Levi-Civita connection.
Unit 5. Introduction to theory of relativity: special theory of relativity, general theory of relativity: Einstein field equations, Schwarzschild’s metric, applications to: redshift effect, Mercury’s perihelion advance, deflection of light about sun.

E. Technology:
Students will be using MAPLE/MATHEMATICA packages and internet resources.

F. Students who may benefit from the course:
Senior Math and Physics students, and graduate students.

G. Follow up courses:
This is a terminal course.

H. Textbook used in the past:
In the past instructors have used “Elements of Differential geometry’’ by R. Milman and G. Parker.